Comparison of Several Multivariate Means

1. Comparison of Several Multivariate Means Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia 1

2. Paired Comparisons • Measurements are recorded under different sets of conditions • See if the responses differ significantly over these sets • Two or more treatments can be administered to the same or similar experimental units • Compare responses to assess the effects of the treatments 2

3. Example 6.1: Effluent Data from Two Labs 3

4. Single Response (Univariate) Case 4

5. Multivariate Extension: Notations 5

6. Result 6.1 6

7. Test of Hypotheses and Confidence Regions 7

8. Example 6.1: Check Measurements from Two Labs 8

9. Experiment Design for Paired Comparisons 1 2 3 n . . . . . . Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random 9

10. Alternative View 10

11. Repeated Measures Design for Comparing Measurements • q treatments are compared with respect to a single response variable • Each subject or experimental unit receives each treatment once over successive periods of time 11

12. 3 4 2 1 Example 6.2: Treatments in an Anesthetics Experiment • 19 dogs were initially given the drug pentobarbitol followed by four treatments Present Halothane Absent Low High CO2 pressure 12

13. Example 6.2: Sleeping-Dog Data 13

14. Contrast Matrix 14

15. Test for Equality of Treatments in a Repeated Measures Design 15

16. Example 6.2: Contrast Matrix 16

17. Example 6.2: Test of Hypotheses 17

18. Example 6.2: Simultaneous Confidence Intervals 18

19. Comparing Mean Vectors from Two Populations • Populations: Sets of experiment settings • Without explicitly controlling for unit-to-unit variability, as in the paired comparison case • Experimental units are randomly assigned to populations • Applicable to a more general collection of experimental units 19

20. Assumptions Concerning the Structure of Data 20

21. Pooled Estimate of Population Covariance Matrix 21

22. Result 6.2 22

23. Proof of Result 6.2 23

24. Wishart Distribution 24

25. Test of Hypothesis 25

26. Example 6.3: Comparison of Soaps Manufactured in Two Ways 26

27. Example 6.3 27

28. Result 6.3: Simultaneous Confidence Intervals 28

29. Example 6.4: Electrical Usage of Homeowners with and without ACs 29

30. Example 6.4: Electrical Usage of Homeowners with and without ACs 30

31. Example 6.4: 95% Confidence Ellipse 31

32. Bonferroni Simultaneous Confidence Intervals 32

33. Result 6.4 33

34. Proof of Result 6.4 34

35. Remark 35

36. Example 6.5 36

37. Multivariate Behrens-Fisher Problem • Test H0: m1-m2=0 • Population covariance matrices are unequal • Sample sizes are not large • Populations are multivariate normal • Both sizes are greater than the number of variables 37

38. Approximation of T2 Distribution 38

39. Confidence Region 39

40. Example 6.6 • Example 6.4 data 40

41. Example 6.10: Nursing Home Data • Nursing homes can be classified by the owners: private (271), non-profit (138), government (107) • Costs: nursing labor, dietary labor, plant operation and maintenance labor, housekeeping and laundry labor • To investigate the effects of ownership on costs 41

42. One-Way MANOVA 42

43. Assumptions about the Data 43

44. Univariate ANOVA 44

45. Univariate ANOVA 45

46. Univariate ANOVA 46

47. Univariate ANOVA 47

48. Concept of Degrees of Freedom 48

49. Concept of Degrees of Freedom 49

50. Examples 6.7 & 6.8 50